| L(s) = 1 | + 3-s − 1.78·5-s + 2.80·7-s + 9-s − 5.53·11-s + 6.04·13-s − 1.78·15-s − 5.94·17-s + 2.45·19-s + 2.80·21-s + 4.94·23-s − 1.81·25-s + 27-s + 2.34·29-s − 1.32·31-s − 5.53·33-s − 5.00·35-s + 5.12·37-s + 6.04·39-s + 1.33·41-s + 3.76·43-s − 1.78·45-s − 0.129·47-s + 0.856·49-s − 5.94·51-s − 2.18·53-s + 9.87·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.798·5-s + 1.05·7-s + 0.333·9-s − 1.66·11-s + 1.67·13-s − 0.460·15-s − 1.44·17-s + 0.563·19-s + 0.611·21-s + 1.03·23-s − 0.362·25-s + 0.192·27-s + 0.435·29-s − 0.238·31-s − 0.962·33-s − 0.845·35-s + 0.843·37-s + 0.967·39-s + 0.208·41-s + 0.574·43-s − 0.266·45-s − 0.0188·47-s + 0.122·49-s − 0.832·51-s − 0.299·53-s + 1.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298148513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298148513\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 139 | \( 1 + T \) |
| good | 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 6.04T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 + 0.129T + 47T^{2} \) |
| 53 | \( 1 + 2.18T + 53T^{2} \) |
| 59 | \( 1 + 1.20T + 59T^{2} \) |
| 61 | \( 1 - 2.86T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 4.62T + 73T^{2} \) |
| 79 | \( 1 + 8.73T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 6.66T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.944901732981231365898941625155, −7.61407788329087741408118705502, −6.73503856805261507484818978044, −5.81056659784345449159094594595, −4.96625634354672295148014557855, −4.39720400232988075843859405925, −3.59063786680559526925037268120, −2.76519064774603181470031156900, −1.90821250819688851130652191159, −0.76417894412303483008988071632,
0.76417894412303483008988071632, 1.90821250819688851130652191159, 2.76519064774603181470031156900, 3.59063786680559526925037268120, 4.39720400232988075843859405925, 4.96625634354672295148014557855, 5.81056659784345449159094594595, 6.73503856805261507484818978044, 7.61407788329087741408118705502, 7.944901732981231365898941625155
